a two-dimensional self-adaptive hp finite element method ...waveguide discontinuities, i.e., the...

A Two-Dimensional Self-Adaptive hp Finite

Element Method for the Characterization of

Waveguide Discontinuities. Part II:

Energy-Norm Based Automatic hp-Adaptivity

Luis E. Garćıa-Castillo 1,2

Departamento de Teoŕıa de la Señal y Comunicaciones. Universidad Carlos III deMadrid. Escuela Politécnica Superior (Edificio Torres Quevedo). Avda. de la

Universidad, 30. 28911 Leganés (Madrid) Spain. Fax:+34-91-6248749

David Pardo

ICES, University of Texas at Austin, Austin TX 78712, USA

Ignacio Gómez-Revuelto

Departamento de Ingenieŕıa Audiovisual y Comunicaciones. UniversidadPolitécnica de Madrid, Madrid, Spain.

Leszek F. Demkowicz

ICES, University of Texas at Austin, Austin TX 78712, USA

Abstract

This is the second of a series of three papers analyzing different types of rectangu-lar waveguide discontinuities by using a fully automatic hp-adaptive finite elementmethod.

In this paper, a fully automatic energy-norm based hp-adaptive Finite Element(FE) strategy applied to a number of relevant waveguide structures, is presented.The methodology produces exponential convergence rates in terms of the energy-norm error of the solution against the problem size (number of degrees of freedom).

Extensive numerical results demonstrate the suitability of the hp-method for solv-ing different rectangular waveguide discontinuities. These results illustrate the flex-ibility, reliability, and high-accuracy of the method.

The self-adaptive hp-FEM provides similar (sometimes more) accurate resultsthan those obtained with semi-analytical techniques such as the Mode Matchingmethod, for problems where semi-analytical methods can be applied. At the sametime, the hp-FEM provides the flexibility of modeling more complex waveguide

Preprint submitted to Elsevier Science 1st March 2006

structures and including the effects of dielectrics, metallic screws, round corners,etc., which cannot be easily considered when using semi-analytical techniques.

Key words: Finite Element Method, hp-adaptivity, Energy-norm, RectangularWaveguides, Waveguide Discontinuities, S-Parameters

1 Introduction

As it was mentioned in the first paper on this series [1], the accurate analy-sis and characterization of “waveguide discontinuities” is an important issuein microwave engineering (see e.g., [2], [3]). Waveguide discontinuities, i.e.,the interruption in the translational symmetry of the waveguide, may be aunavoidable result of mechanical defects or electrical transition in waveguidesystems, or they may be deliberately introduced in the waveguide to performa certain electrical function. Specifically, discontinuities in rectangular waveg-uide technology are very common in the communication systems working inthe upper microwave and millimeter wave frequency bands. In many cases,the rectangular waveguide discontinuities can be analyzed in two-dimensions(2D) because of the invariant nature of the geometry along one direction. Thisis the case of the so called H-plane and E-plane rectangular waveguide dis-continuities, which are the target of this work. It is worth noting that a largenumber of structures and devices fit into this category.

In this paper, a fully automatic energy-norm based hp-adaptive Finite Element(FE) strategy [4,5], which has been extended for electromagnetic applications[6,7,8,9,10,11,12,5] is applied to a number of relevant waveguide structures.The adaptive methodology has a number of advantages that makes it suitablefor the analysis of complex structures (containing several waveguide sections,discontinuities, complex geometries, dielectrics, etc.) in contrast to other semi-analytic and numerical techniques. Namely:

• It automatically resolves different types of singularities i.e., different types

Email addresses: [email protected], [email protected] (Luis E.Garćıa-Castillo), [email protected] (Ignacio Gómez-Revuelto),[email protected] (Leszek F. Demkowicz).1 This work has been initiated during a stay of the first author at ICES supported bythe Secretaŕıa de Estado de Educación y Universidades of Ministerio de Educación,Cultura y Deporte of Spain. The authors wants also to acknowledge the support ofMinisterio de Educación y Ciencia of Spain under project TEC2004-06252/TCM.2 On leave from Departamento de Teoŕıa de la Señal y Comunicaciones. Universidadde Alcalá, Madrid, Spain

2

of discontinuities.• It efficiently deals with high frequencies, that is, it delivers a low dispersion

error [13,14].• It provides high-accuracy results, so the S-parameters (see comments below)

[1] can be accurately computed.• It allows for modeling of complex (non-uniform) geometries.

The waveguide theory and a detailed analysis of rectangular waveguide discon-tinuities (including the finite element variational formulations used) were pre-sented in the first part of the series [1]. Extensive numerical results presentedin this part illustrate the flexibility, reliability, and high-accuracy simulationsobtained with this methodology, providing more accurate results than withsemi-analytical (in particular, Mode-Matching —MM— [15], [16, Chapter 9])techniques. The adaptive methodology is shown to produce exponential con-vergence rates in terms of the energy-norm error of the solution against theproblem size (number of degrees of freedom). Thus, the electromagnetic fieldis accurately known (with a user pre-specified degree of accuracy) inside thestructure. The high accuracy is essential in the microwave engineering designaiming at finding optimum location and size of tuning elements (e.g., screws,dielectric posts, etc.) as well as for an a posteriori analysis of such structures.

The presented numerical results include computation of the scattering pa-rameters of the structure, which are widely used in microwave engineeringfor the characterization of microwave devices. The notion of the scatteringparameters (or S-parameters), and their computation using a finite elementsolution, have been explained in the first paper of the series [1]. As the quan-tities of interest for the microwave engineer are mainly the S-parameters, agoal-oriented approach (in terms of the S-parameters) may be also desirable.In the third paper of the series [17], results obtained using hp energy-normadaptivity are compared against those using a goal-oriented hp-adaptivity ap-proach [18,19,20]. Results show that both methods are suitable for simulationof waveguide discontinuities.

The organization of the paper is as follows. The hp finite element discretizationand automatic adaptivity strategy are briefly described in Section 2.1 and2.2, respectively. The refinement strategy is based on the minimization of theprojection based interpolation error, which is defined in Section 2.2.1. The stepsof the mesh optimization algorithm are described in Section 2.2.2. Extensivenumerical results, both for the E-plane and H-plane simulations, are shown inSection 3. Finally, some conclusions are given in Section 4.

3

2 hp Finite Elements and Automatic Adaptivity

In order to solve the presented electromagnetic problems, a numerical tech-nique that provides low discretization errors and, simultaneously, solves thediscretized problem without prohibitive computational cost, is needed. In thiscontext, an adaptive hp-Finite Element Method satisfies both properties.

2.1 hp-Finite Elements (FE)

Each finite element is characterized by its size h and order of approximationp. In the h-adaptive version of FE method, element size h may vary fromelement to element, while order of approximation p is fixed (usually p=1,2).In the p-adaptive version of the FE method, p may vary locally, while h re-mains constant throughout the adaptive procedure. Finally, a true hp-adaptiveversion of FE method allows for varying both h and p locally.

The hp-FE method used in this paper utilizes edge (Nédélec) elements ofvariable order of approximation. FE spaces associated to those elements havebeen carefully constructed (see [21] for details) so in combination with theprojection based interpolation operators (defined below), the commutativity ofthe de Rham diagram is guaranteed. This commutativity property is essentialfor showing convergence and stability of the FE method for Electromagnetics[21].

The main motivation for the use of hp-FEM is given by the following result:“an optimal sequence of hp-grids can achieve exponential convergence for ellip-tic problems with a piecewise analytic solution, whereas h- or p-FEM convergeat best algebraically” (see [22,23,24,25,26,27,28]).

Next, the fully automatic hp-adaptive strategy is presented. Given a problemand a discretization tolerance error, the objective is to generate automatically(without any user interaction) an hp-grid that does not exceed the discretiza-tion error tolerance and, at the same time, it employs a minimum number ofdegrees of freedom (d.o.f.), by orchestrating an optimal distribution of elementsize h and polynomial order of approximation p. By doing so, it is possible toachieve exponential convergence rates in terms of the error vs. the number ofd.o.f.

4

2.2 Fully Automatic hp-Adaptivity

The self-adaptive strategy iterates along the following steps. First, a given(coarse) hp-mesh is globally refined both in h and p to yield a fine mesh, i.e.,each element is broken into four element sons (eight in 3D), and the orderof approximation is raised uniformly by one. Then, the problem of interestis solved on the fine mesh. The difference between the fine and coarse gridsolutions is used to guide optimal refinements over the coarse grid. More pre-cisely, the next optimal coarse mesh is then determined by minimizing theprojection based interpolation error of the fine mesh solution with respect tothe optimally refined coarse mesh (see [4,29] for details).

The adaptive strategy is very general, and it applies to H1-, H(curl)-, andH(div)-conforming discretizations. Moreover, since the mesh optimization pro-cess is based on minimizing the interpolation error rather than the residual,the algorithm is problem independent and it can also be applied to nonlinearand eigenvalue problems.

The hp self-adaptive strategy incorporates also a two-grid iterative solver,which allows to solve the fine grid problems efficiently. Indeed, it has beenshown in [30,31] that it is sufficient a partially converged fine grid solutionto guide optimal hp-refinements. Thus, only few two-grid solver iterations areneeded (below ten per grid).

In the remainder of this section, the projection based interpolation operator[32,33], which is the main ingredient of the mesh optimization algorithm, ispresented first. Then, the mesh optimization algorithm is briefly described.

2.2.1 The projection based interpolation operator

The idea of projection based interpolation operator is based on three proper-ties.

• Locality: Determination of element interpolant of a function should involvethe values (and derivatives) of the interpolated function in the element only.

• Conformity: The union of element interpolants should be globally conform-ing.

• Optimality: The interpolation error should behave asymptotically, both inh and p, in the same way as the actual approximation error.

The H1-conforming projection based interpolation operator is presented first.Let u ∈ H1+�(K) with � > 0. Locality and conformity imply that the inter-

5

polant w = Πu should match the interpolated function u at vertexes:

w|vert = u|vert (1)

With the vertex values fixed, we project over each edge, i.e.:

w := arg minv:(v−u)|vert=0

‖ v − u ‖edge (2)

This definition preserves locality and conformity. It also preserves optimalityprovided that the optimal edge norm is selected, which is dictated by the prob-lem being solved and the Trace Theorem (see [21] for details). For example,in 1D, the optimal edge norm is the H10 -norm. In 2D, the H

1/2-seminorm, andin 3D, the L2-norm, should be used.

Using the same argument, once vertex and edge values are fixed, projectionover the interior of the element (faces in 3D) is performed. Thus, the projectionbased interpolation operator for 2D H1-problems is formally defined as:

w(v) := u(v) for each vertex v

|w − u|12

,e→ min for each edge e

|w − u|1,K → min in the interior of element K

(3)

For a definition of projection based interpolation operator for 3D H1-problems,see [34].

Similarly, a projection based interpolation operator can be defined for elementsin H(curl), which is the space of interest for the electromagnetic field. GivenE 3 in H(curl), the projection based interpolator Πcurl specialized to the 2Dcase (for the 3D see [32,35]), is denoted by Ep = ΠcurlE, where Ep is given by:

‖ Ept − Et ‖−12

,e→ min for each edge e

|∇× Ep −∇× E|0,K → min

(Ep − E, ∇φ)0,K = 0, for every “bubble” function, in the interior of element K(4)

3 E is used here to abstractly denote an element in H(curl). In this paper, discretiza-tions in H(curl) are used for the magnetic field on the H-plane and the electric fieldon the E-plane of the structures.

6

Here, the bubble functions come from an appropriate polynomial space mappedby the gradient operator onto the subspace of fields E with zero curl and tan-gential trace on the element boundary.

A similar operator can be defined for H(div) problems (see [21]).

Finally, it is important to mention that the de Rham diagram equipped withthese projection based interpolation operators commutes (see [21,12,32] fordetails), which is critical for proving stability and convergence properties ofthe FEM for Maxwell equations.

2.2.2 The mesh optimization algorithm

The mesh optimization algorithm in 2D follows the next steps.

• Step 0: Compute an estimate of the approximation error on the coarse grid.The approximation error on the coarse grid is estimated by simply com-

puting the norm of the difference between the coarse and the fine grid so-lutions. If the difference (relative to the fine grid solution norm) is smallerthan a requested error tolerance, then the fine mesh solution is delivered asthe final solution, and the process stops.

• Step 1: For each edge in the coarse grid, compute the error decrease ratefor the p refinement, and all possible h-refinements.

Let p1, p2 be the order of the edge sons in the case of h-refinement, andlet E = Eh/2,p+1 denote the fine grid solution. Then, the error decrease rateis computed as:

Error decrease (ĥp) =‖Eh/2,p+1 −Πcurlhp E‖ − ‖Eh/2,p+1 −Πcurlĥp E‖

(p1 + p2 − p),

where ĥp = (ĥ, p̂) is such that ĥ ∈ {h, h/2}. If ĥ = h, then p̂ = p + 1. Ifĥ = h/2, then p̂ = (p1, p2), where p1 + p2 − p > 0, max{p1, p2} ≤ p + 1.

• Step 2: For each edge in the coarse mesh, choose between p and h refinement,and determine the guaranteed edge error decrease rate.

The optimal refinement is found by comparing the error decrease corre-sponding to the p-refinement with all competitive h-refinements. Compet-itive h-refinements are those that result in the same increase in the num-ber of degrees-of-freedom (d.o.f.) as the p-refinement, i.e., ĥ = h/2 andp1 + p2 − p = 1.

Next, the guaranteed rate with which the interpolation error must de-crease over the edge is determined. That is, for each edge the maximumof the error decrease rates for the p-refined edge and all possible h-refinededges is computed.

7

• Step 3: Select edges to be refined.Given the guaranteed rate for each edge in the mesh, the maximum rate

for all elements is calculated

guaranteed ratemax = maxe (edge e guaranteed rate) .

All edges that produce a rate within 1/3 of the maximum guaranteedrate, are selected for a refinement. The factor 1/3 is somehow arbitrary.

• Step 4: Perform the requested h-refinements enforcing the 1-irregularity ruleof the mesh.

A loop through elements of the coarse grid is performed. If at least oneedge of the element is to be broken, the element is refined accordingly. Asin [4,29], element isotropy flags are computed. Isotropic h-refinement areenforced if the error function within the element changes comparably inboth element directions.

After this step, the topology of the new coarse mesh has been determined,and it remains only to establish the optimal distribution of orders of ap-proximation for the involuntarily h-refined edges, and for the interior nodesof the element i.e., those nodes that are not located on the boundary ofthe element. For the interior nodes, the starting point for the minimizationprocedure will be based on the order of approximation p for the adjacentedges 4 .

• Step 5: Determine the optimal orders of approximation p for the refinededges and elements.

This step consists basically of p-adaptivity over a given grid with thefine grid as a reference solution. Unfortunately, due to the possible presenceof involuntary edge h-refinements and too low p for h-refined elements, theinterpolation error of the coarse grid after step 4 may actually be larger thanthe interpolation error for the original coarse mesh. Thus, extra technicaldetails are considered in order to guarantee interpolation error decrease.These details are quite involving, and are described in [12].

Some remarks on the mesh optimization algorithm follow:

• A similar but yet more involved mesh optimization algorithm has beenimplemented for 3D problems, although the 3D electromagnetic version isstill under development.

• The main difference between the fully automatic hp-adaptive strategy for

4 For triangles, the initial order of approximation will be equal to the maximumof the three edges of the element. For quadrilaterals, we have a horizontal and avertical order of approximation p = (ph, pv). In this case, the starting point for theminimization procedure will be the maximum of the two horizontal edges for ph andthe maximum of the two vertical edges for pv.

8

elliptic and electromagnetic problems resides in the definition of the projec-tion based interpolation operator.

• A similar algorithm can be implemented for H(div) problems.

3 Numerical Results

In the following, a number of rectangular H-plane and E-plane waveguidediscontinuities, as well as more complex structures obtained by combiningseveral discontinuities, are analyzed. The analysis of all these structures isperformed by using the fully automatic hp-adaptive FE strategy presentedabove.

TE10 mode excitation has been used in all the structures. Also, the ratio ofthe broad dimension a to the narrow dimension b of the rectangular waveg-uide sections is considered to be a/b = 2. The results correspond to a givenfrequency which is chosen to be in the middle of the monomode region, i.e.,k = 1.5Kc10. Exceptionally, the structure analyzed in Section 3.2.4 is solvedfor a large number of frequencies within a given frequency region, in order tocharacterize its frequency response. The lengths of the waveguide sections thatconnect the discontinuity to the ports of the structure is typically around onewavelength for the H-plane structures, and around half a wavelength for theE-plane structures. This is enough for the first absorbing boundary conditionused at the ports to perform correctly.

Typically, quite coarse meshes are used as initial grids in order to assess therobustness of the hp strategy in the context of real engineering analysis inwhich the initial mesh has to be as coarse as possible in order to simplify themesh generation process. The convergence history is always shown using a logscale for the energy error (in percent of the energy norm) in the ordinate axis

and a scale corresponding to N1/3dof (being Ndof the number of degrees of freedom

in the mesh) in the abscissas axis. Thus, according to [27] and referencestherein, an straight line should appear in the plot showing the theoreticalexponential convergence that can be achieved with an optimal hp adaptivitystrategy. Note that the abscissas scale corresponds to N

1/3dof while abscissas axis

tics should be read as Ndof in the plots.

The scattering parameters obtained using the hp-FEM are compared withvalues computed with the Mode Matching (MM) method (see e.g., [15], [16,Chapter 9]). The MM method can be considered as a semi-analytic method. Itconsists of the decomposition of the domain of the problem into several simpledomains, typically with translational symmetry, in which, an analytical modalexpansion can be performed. Imposing the tangential continuity of the fieldand orthogonality of the modes yields a system of equations in which the

9

unknowns are the coefficients of the modal expansions.

The FEM scattering parameter results delivered from the hp adaptivity aremore accurate than MM results. In this context, it is important to pointout that MM results are typically considered as a reference for the engineer-ing analysis of discontinuities in rectangular waveguide technology, as for thestructures shown below. In addition, the hp-FE technology allows for modelingof more complex structures which cannot be solved using the MM.

3.1 H-plane discontinuities

The analysis of several H-plane discontinuities is considered next. The bound-ary condition of the metallic conductors represents a Neumann boundary con-dition for the H-plane formulation.

3.1.1 H-plane waveguide section

The first structure shown in Fig. 1 is a simple rectangular waveguide section.This structure is chosen as a first verification of the code, specifically for theboundary conditions at the ports and the control of the dispersion error. Sincethere is no discontinuity in the translational symmetry for this structure, itmay be analyzed by means of either the H-plane or E-plane formulations. Theresults shown in this section correspond to the H-plane analysis. Also, becausethere is no discontinuity, the scattering parameters of the structure are knownto be S11 = S22 = 0 and S21 = S12 = exp(−jβ10l), where l denotes thewaveguide section length. In this case, l is equal to 2 wavelengths and, thus,S21 = S12 = exp(−j4π) = 1. The field solution is also known: it correspondsto the field TE10 mode inside of the waveguide section.

The solution is smooth: a half-sine type variation in the y direction (the ±ξlocal axis of the waveguide) and constant in amplitude and phase variation asexp(−jβ10x) along the x direction (the ±ζ local axis of the waveguide). Thus,the hp-adaptive strategy is expected to deliver an increase in the polynomialorder of approximation p. The initial mesh used for the analysis is shown inFig. 2 together with some intermediate meshes. The colors indicate, accordingto the scale on the right, the order p of the elements. It is important to notethat the order corresponds to the H1 Lagrange multiplier and that the field ofH(curl) is of order p− 1. As an example, the green color of the initial mesh ofFig. 2(a) indicate that all elements are of order 3 for the Lagrange multiplierand order 2 for the magnetic field. It is observed that order p is increaseduntil the maximum p (p = 9) is reached in the fine grid; from this moment, hrefinement is selected until the specified error criterion is satisfied.

10

The convergence history for the exact error and the estimated error is plot-ted in Fig. 4, showing the quality of the error estimation and the exponentialbehavior of the error. The exponential convergence is deduced from the ob-servance of a straight line in the plot for the log type ordinates axis and theN

1/3dof scale in the abscissas axis set in the figure. It is worth noting that the

slope change in the convergence history corresponds to the moment when themaximum p is reached, so h refinements are forced. In other words, this slow-down in convergence would not have occurred if higher order elements wereallowed. A plot of the field in the structure, specifically, |Hy|, is shown inFig. 3 where it is clearly observed the sine and the zero variations along they and x axis, respectively. Note that the the zero variation along the x axisis because we are referring to |Hy| and not Hy. A constant magnitude in thedirection of propagation means that there is only one wave in the waveguideand, thus, |Hy| = |H in0 | | sin(πξ/a)|, so it does not depend on the ζ ≡ ±xdirection. If a discontinuity had generated a reflected wave, a stationary wavepattern would have been observed at the input waveguide (as it is seen in theexamples below).

The scattering parameters have been computed at each iteration step of thehp strategy. Due to the reciprocity and symmetry of the structure, the scatter-ing behavior of the discontinuity is characterized by performing one analysis(exciting any of the two ports).The results for the first iterations are shownin Tab. 1. A fast convergence is observed.

Table 1Scattering parameters for the H-plane waveguide section

|S11| |S21| arg(S21)

Iter. 1 1.0302e-02 0.9991183 10.1204◦

Iter. 2 5.9652e-04 0.9994050 0.9797◦

Iter. 3 4.6872e-07 0.9999995 0.0402◦

Iter. 4 2.7117e-07 0.9999997 0.0013◦

Analytic 0.0 1.0 0.0◦

11

y

x

x

H-plane

l

Port 2

y

ζPort 1

ξ

η

ξζ

η

ε0, µ0b

ε0, µ0Ω

Γ2pΓ1p

ΓN

H-planeζ

ξζ

ξ

a

l

ΓN

a

Figure 1. H-plane waveguide section

12

x

y

z

(a) Initial mesh

11 12 13

x

y

z

(b) Mesh 4th iteration

14 15

16

17

28 29

30

31

42 43

44

45

x

y

z

(c) Mesh 5th iteration

58 59

60

61

70 71

72

73

82 83

84

85

94 95

96

97

110 111

112

113

122 123

124

125

134 135

136

137

146 147

148

149

162 163

164

165

174 175

176

177

188 189

190

191

202 203

204

205

x

y

z

(d) Mesh 7th iteration

Figure 2. Initial mesh and some hp meshes for the H-plane waveguide section

Figure 3. Magnitude of Hy, i.e., |Hy|, corresponding to the H-plane waveguide sec-tion

13

-5

-4

-3

-2

-1

0

1

2

228 448 778 1239 1854 2645 3634 4843 6294

log1

0 (r

elat

ive

erro

r %

)

Ndof

H-plane waveguide section

exact-energy-errorestimate-energy-error

Figure 4. Convergence history for the H-plane waveguide section (energy norm errorfor the magnetic field solution)

14

3.1.2 H-plane right angle bend

An H-plane 90◦ bend is analyzed next. The structure is shown in Fig. 5. Thestructure is analyzed by exciting port 1 (on the left). The bend is obviously acommon part of microwave circuits. The initial mesh used for the analysis isshown in Fig. 6. Despite the coarseness of this mesh, the hp-strategy achievesan energy error lower than 1% error after 5 iterations. The convergence his-tory (up to an error as low as 0.01%) is shown in Fig. 9. The final mesh isshown in Fig. 8. It is observed how the mesh is refined around the corner. Thehp-strategy in this case tends also to increase the p. Actually, all elements ofthe final mesh (except those near the corner) have reached the maximum porder. This is the right strategy as the solution of the problem is smooth (theboundary condition at the conductors for the H-plane formulation is of ho-mogeneous Neumann type 5 ). The h-refinement around the corner is preciselydue to the fact that the maximum p has been reached and, in order, to reducethe error in this region, the elements must be made smaller.

A plot of the field in the structure, specifically, |Hy|, is shown in Fig. 7. They-component corresponds to the local ξ component at the excitation portand the local ζ component at the transmitted port. Notice the figure thestationary wave pattern in the input waveguide (between the excitation portand the bend) because of the combination of the two waves propagating inopposite directions (the excited wave and the reflected wave at the bend). Nostationary wave is observed in the output waveguide as there is only one wavepropagating outward the transmitted port. As in the previous case, S21 = S12and S22 = S11. The results for S11 and S21 are shown (for some of the hpmeshes) in Tab. 2. The scattering parameters computed with the hp-FEMmethod are compared with those obtained with a MM technique. Only foursignificant digits are shown in the table as the MM results are presumed tohave no more tan 4 digits of accuracy. Observe the very good agreement ofthe hp-FEM results with those provided by MM; better than 1% after thesecond iteration. After the fourth/fifth iteration, the FEM results seem to bemore accurate than those provided by the MM, as implied by the convergencepattern shown in the table.

5 The same domain but with Dirichlet boundary conditions corresponds to theE-plane bend which is analyzed in Section 3.2.1.

15

xy

y

x

H-plane

Port 1ξ

η

η

ξ

Port 2

b

l1

b

l2

aa

ζ

ζ

ε0, µ0

ε0, µ0Ω

ΓN

ΓNH-plane

ζ

ξ

ξ

Γ2p

l1

l2

a

ζ

Γ1p

a

Figure 5. H-plane 90 degrees bend

16

11 12

13

x

y

z

Figure 6. Initial mesh for the H-plane 90◦ bend

Figure 7. Magnitude of Hy, i.e., |Hy|, corresponding to the H-plane 90◦ bend

17

14

68

100

132

164

196

197

198

199

166

167

134

135

102

103

70

71

16

17

206

207

208

209

175

176

177

143

144

145

111

112

113

79

80

81

43

44

45

28 29

30

58 59

60

90

91

92

122

123

124

154

155

156

186

187

188

218

219

220

221

x

y

z

Figure 8. Final hp mesh for the H-plane 90◦ bend

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

208 387 647 1003 1471 2066 2802 3695 4760

log

(rel

ativ

e er

ror

%)

Ndof

H-plane 90 degrees Bend

energy error

Figure 9. Convergence history for the H-plane 90◦ bend

18

Table 2Scattering parameters for the H-plane 90◦ bend

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.5465 0.8372 10.049◦ 112.615◦

Iter. 2 0.4459 0.8951 16.948◦ 101.604◦

Iter. 3 0.4148 0.9099 5.639◦ 95.665◦

Iter. 4 0.4156 0.9096 5.619◦ 95.617◦

MM 0.4161 0.9093 5.345◦ 95.345◦

19

3.1.3 H-plane symmetric inductive iris

The structure is shown in Fig. 10. The discontinuity consists of the narrowingof the broad dimension of the waveguide along a certain length. This region isreferred to as an iris 6 . Because the iris is centered with respect the waveguidebroad dimension, it is called a symmetric iris. Finally, the term inductive isused because the discontinuity scattering behavior with respect to the planesof the discontinuity is equivalent to an inductance. The character of the dis-continuity, inductive or capacitive, can be deduced by observing which fieldlines (electric or magnetic) of the waveguide mode (the TE10 in this case) are“cut” by the discontinuity [2]. It is clear from Fig. 10 that only the magneticfield lines are cut in this case (the electric field of the TE10 is perpendicularto the H-plane of the waveguide). Thus, the symmetric iris of Fig. 10 is of theinductive type. An analogous structure, but of capacitive type, is consideredin Section 3.2.3.

The initial mesh used for the analysis is shown in Fig. 11. The analysis ismade by exciting port 1 (at the left). The convergence history (up to an erroras low as 0.02%) is shown in Fig. 12. Notice the exponential convergence ofthe method.

A plot of the field in the structure, specifically, |Hy|, is shown in Fig. 13.The y-component corresponds to the ±ξ component of the field modes in thewaveguide. Observe the stationary wave pattern at the input port due to thewave reflected from the discontinuity and a singular behavior of the field atthe re-entrant corners. The magnitude of the fields is higher at the left corners.This is “caught” by the hp strategy that refines around the left corners duringthe first few iterations, and once the error around the left corners is controlled(comparable to other regions of the structure), it starts to “see” the errorcorresponding to the region around the right corners (see figures 14 and 15).

As in the previous case, S21 = S12 and S22 = S11. The results for S11 and S21are shown (for some of the hp meshes) in Tab. 3. Only the results of the firstiterations are shown as the hp FEM results for the consecutive meshes arepresumed to be more accurate than those of MM. Observations analogous tothose mentioned in the previous case may be made here.

6 The term iris is used in this context to refer to an aperture that connects twowaveguide sections.

20

y

x

a

x

H-plane

Port 2

y

ζPort 1

ξ

η

ξζ

η

Ω

Γ2pΓ1p

H-planeζ

ξζ

ξ

ΓN

a

b

l1l

l2

t

ε0, µ0

ε0, µ0

l

ΓN

l1 t l2

Figure 10. H-plane symmetric inductive iris (l/a = 0.6, t/a = 0.2)

21

x

y

z

Figure 11. Initial mesh for the H-plane symmetric inductive iris

-2

-1.5

-1

-0.5

0

0.5

1

1.5

1035 1864 3046 4645 6725 9347 12575 16473 21102

log

(rel

ativ

e er

ror

%)

Ndof

H-plane iris inductive symmetric

energy-error

Figure 12. Convergence history for the H-plane inductive iris

22

Figure 13. Magnitude of Hy for the H-plane inductive iris

x

y

z

Figure 14. 11th mesh for the H-plane symmetric inductive iris showing the refine-ments around the left corners

23

x

y

z

Figure 15. 19th mesh for the H-plane symmetric inductive iris showing the refine-ments around the left and also right corners

Table 3Scattering parameters for the H-plane symmetric inductive iris

|S11| |S21| arg(S11) arg(S21)

Iter. 2 0.7333 0.6799 -157.92◦ -27.243◦

Iter. 4 0.7401 0.6725 -156.59◦ -26.325◦

Iter. 6 0.7386 0.6741 -156.48◦ -26.588◦

Iter. 8 0.7414 0.6711 -156.46◦ -26.287◦

MM 0.7417 0.6708 -156.51◦ -26.259◦

24

3.1.4 H-plane zero thickness septum

The structure, shown in Fig. 16, consists of an obstacle (the septum) placed atthe center of the waveguide. The obstacle exhibits a translational symmetryalong the narrow dimension of the waveguide (η local axis of the waveguideports). Thus, it can be analyzed using the H-plane formulation. Also, as forthe previous structure, the scattering behavior is basically inductive since theonly field lines that are cut by the septum are those of the magnetic field.

The initial mesh used for the analysis is shown in Fig. 17. The analysis is madeby exciting port 1 (on the left). The convergence history is shown in Fig. 18.It is observed that the error convergence (except for the first couple of meshes,due to a deliberate coarseness of the initial mesh) behaves as predicted by thetheory and a straight line is obtained in the plot (which means exponentialconvergence).

An example of one of the hp meshes provided by the adaptivity, specifically,the mesh of the 7th iteration, is shown in Fig. 19. Again, as predicted by thetheory, it is observed the h-refinements towards the corners where there is asingular behavior of the field and the p-refinements in the regions where thefield variation is smooth.

A plot of |Hy| is shown in Fig. 20. The y-component corresponds, as in theprevious case, to the ±ξ component of the field modes in the waveguide.Observe the stationary wave pattern at the input port due to the wave reflectedfrom the discontinuity and the singular behavior of the field at the septumcorners.

The results for S11 and S21 corresponding to some of the iterations of thehp adaptivity are shown in Tab. 4. As in the previous cases, S21 = S12 andS22 = S11 due to reciprocity and symmetry. Only the results up to the 10thiteration are shown, since the error in the scattering parameters obtained fromthe hp meshes (for iterations higher than the 10th) is expected to be lowerthan the one of the MM results.

25

y

x

a

x

H-plane

Port 2

y

ζPort 1

ξ

η

ξζ

η

Ω

Γ2pΓ1p

H-planeζ

ξζ

ξ

ΓN

a

b

l1

l2

ε0, µ0

ΓN

l1 l2

lε0, µ0

l

ΓN

Figure 16. H-plane zero thickness septum (l/a = 0.1)

26

x

y

z

Figure 17. Initial mesh for the H-plane zero thickness septum

-1.5

-1

-0.5

0

0.5

1

1.5

566 984 1569 2351 3355 4612 6149 7995 10177

log

(rel

ativ

e er

ror

%)

Ndof

H-plane zero thickness septum

energy-error

Figure 18. Convergence history for the H-plane zero thickness septum

27

x

y

z

Figure 19. 7th mesh for the zero thickness septum

Figure 20. Magnitude of Hy corresponding to the H-plane zero thickness septumshowing a stationary wave pattern at the input waveguide and singular behavior ofthe field at the septum corners

28

Table 4Scattering parameters for the H-plane zero thickness septum

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.7383 0.6743 206.84◦ -53.120◦

Iter. 4 0.7466 0.6653 205.58◦ -53.801◦

Iter. 7 0.7740 0.6332 208.64◦ -51.486◦

Iter. 10 0.7785 0.6277 209.03◦ -50.962◦

MM 0.7788 0.6273 209.03◦ -50.939◦

29

3.1.5 H-plane zero length septum

This discontinuity is also a septum but it is placed transverse to the wavepropagation. The dimension of the septum along the propagation direction isconsidered zero. A representation of the structure is shown in Fig. 21.

The analysis is made by exciting port 1 (on the left) and considering thecoarse initial mesh of Fig. 22. As in the other septum discontinuity, the errorconvergence (see Fig. 23) behaves as predicted by the theory (exponentialconvergence) and a straight line is obtained in the plot. Fig. 24 shows, as anexample, the mesh corresponding to the 7th iteration where the refinementaround the septum corners can be seen. This is what is expected from the fieldsolution. The y component of the solution (its magnitude) is shown in Fig. 25.Again, a stationary wave pattern is observed in the input waveguide section.

Finally, the results for S11 and S21 corresponding to some of the iterations ofthe hp adaptivity are shown in Tab. 5. Only results up to the 11th iterationare shown, since the error in the scattering parameters obtained from the hpmeshes is expected to be lower (for iterations higher than the 11th) than theone of the MM results.

Table 5Scattering parameters for the H-plane zero length septum

|S11| |S21| arg(S11) arg(S21)

Iter. 2 0.7200 0.6939 50.520◦ -37.987◦

Iter. 5 0.7817 0.6236 56.432◦ -33.550◦

Iter. 8 0.7883 0.6153 57.042◦ -32.962◦

Iter. 11 0.7896 0.6136 57.165◦ -32.840◦

MM 0.7897 0.6135 57.171◦ -32.829◦

30

y

x

a

a

x

H-plane

Port 2

y

ζPort 1

ξ

η

ξζ

η

Ω

Γ2pΓ1p

H-planeζ

ξζ

ξ

ΓN

b

l1

l2

ΓN

l1 l2

ε0, µ0

ε0, µ0

ΓN

t

t

Figure 21. H-plane zero length septum (t/a = 0.1)

31

x

y

z

Figure 22. Initial mesh for the H-plane zero length septum

-2

-1.5

-1

-0.5

0

0.5

1

1.5

795 1555 2689 4272 6379 9086 12466 16596 21549

log

(rel

ativ

e er

ror

%)

Ndof

H-plane zero length septum

energy-error

Figure 23. Convergence history for the H-plane zero length septum

32

x

y

z

Figure 24. 7th mesh for the zero length septum

Figure 25. Magnitude of Hy corresponding to the H-plane zero length septum show-ing a stationary wave pattern at the input waveguide and singular behavior of thefield at the septum corners

33

3.2 E-plane discontinuities

The analysis of several E-plane discontinuities is considered next. In contrastto the H-plane formulation, the boundary condition of the metallic conductorsis of the Dirichlet type for the E-plane formulation.

3.2.1 E-plane right angle bend

This discontinuity (Fig. 26) is as the one of Section 3.1.2, a 90 degrees bend.However, the plane of the bend in this case is the E-plane. The domain shapeis the same as for the H-plane bend (actually, the initial mesh is also the same,see Fig. 6—) but, this time, the homogeneous Dirichlet boundary conditionat the conductors are employed. This, apparently, produces a field singular-ity that occurs at the corner. The hp-strategy behaves as expected and, incontrast to the H-plane bend case, an h-refinement toward the singularity isobserved while increasing the p backward. One of the meshes obtained by thehp-adaptivity procedure is shown in Fig. 27.

Plots of the field component magnitudes |Ey| and |Ex| are shown in Fig. 28. Astationary wave pattern is observed in the input waveguide (between the ex-citation port and the bend). The y-component corresponds to the componentalong the local −η axis of the excitation port and the component along theζ local axis of the transmitted port. Since the TE10 does not have ζ compo-nent, the Ey component is null (numerically null provided that the port is farenough from the discontinuity) at the transmitted port (port 2). Analogously,the Ex component is null at the incident port.

The convergence history is shown in Fig. 29. Except for the peak aroundthe third hp iteration (due to the coarseness of the initial mesh), the errorshows an exponential decay. With respect to the convergence of the scatteringparameters, Tab. 6 shows their values (in magnitude and phase) for some ofthe iterations. The MM results are shown for comparison purposes. A goodagreement is observed. It is worth noting again that only the results up to the10th iteration are shown, since the error in the scattering parameters obtainedfrom the hp meshes (for iterations higher than the 10th) is lower than the onecoming from the MM results.

34

y

x

xy

ηζ

ξ

ε0, µ0Ω

Γ2p

l1

l2

Γ1p

η

ξζ

l2

Port 1

ε0, µ0

Port 2

E-plane

b

b

ΓD

ΓD

η

ζ

ζ

η

E-plane

bb

l1

a

a

Figure 26. E-plane 90 degrees bend

35

x

y

z

Figure 27. hp mesh of 11th iteration of the E-plane 90◦ bend

Table 6Scattering parameters for the E-plane 90◦ bend

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.5542 0.8323 -47.810◦ -137.81◦

Iter. 4 0.5387 0.8425 -46.794◦ -136.94◦

Iter. 7 0.5487 0.8360 -48.590◦ -138.45◦

Iter. 10 0.5499 0.8352 -48.558◦ -138.49◦

MM 0.5507 0.8347 -48.462◦ -138.46◦

36

(a) |Ey|

(b) |Ex|

Figure 28. Magnitudes of Ey and Ex corresponding to the E-plane 90◦ bend showinga singular behavior of the field at the corner

37

-8

-7

-6

-5

-4

-3

-2

-1

4 5 6 7 8 9 10 11 12 13 14 15

log

(rel

ativ

e er

ror

%)

Ndof1/3

E-plane 90 degrees Bend

energy-error

Figure 29. Convergence history for the E-plane 90◦ bend

38

3.2.2 E-plane right angle bend with a round corner

This structure is also a 90◦ bend in the E-plane but with a round corner(see Fig. 30). In practice, depending on the mechanical process used to buildthe bends, the bends may either have sharp corners or round corners (as inthis case). Although there is no field singularity because of the roundnessof the corner, there is a high variation of the fields around the corner, andthe adaptivity behaves analogously to the case of a sharp corner (in the pre-asymptotic regime, as a high variation in the fields is “seen” as a singularity).

The analysis is made by exciting port 1 (on the left) and considering the coarseinitial mesh of Fig. 31. Fig. 32 shows a sample mesh corresponding to the 10thiteration. A refinement pattern similar to the one of Fig. 27 can be observed.The exponential convergence history is shown in Fig. 34.

Plots of the field component magnitudes |Ey| and |Ex| are shown in Fig. 33.Comments analogous to those made on the E-plane bend with a sharp arevalid for this case as well. Tab. 7 shows the values (in magnitude and phase)of S11 and S21 for the first few iterations. A fast convergence is observed andseven digits are needed in order to be able to observe the convergence of thescattering parameters. No MM results are shown for this case. The analysisof this structure by MM requires the use of special functions and somehowdiffers of what it is usually referred to as the MM method.

Table 7Scattering parameters for the E-plane 90◦ bend with round corner

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.5835441 0.8120788 -86.29061◦ -176.29024◦

Iter. 2 0.5835960 0.8120416 -86.27239◦ -176.27328◦

Iter. 3 0.5836238 0.8120215 -86.26833◦ -176.26855◦

Iter. 4 0.5836237 0.8120217 -86.26894◦ -176.26880◦

39

y

x

xy

Ω

Γ2p

Γ1p

η

ξζ

l2

Port 1

ε0, µ0

Port 2

E-plane

ΓD

η

ζ

ζ

η

E-plane

l1

a

a

b

b

ε0, µ0

b b

l2

l1

ΓD r

r ηζ

ξ

Figure 30. E-plane 90 degrees bend with round corner (r/b = 0.2)

40

x

y

z

Figure 31. Initial mesh for the E-plane 90◦ bend with round corner

x

y

z

Figure 32. 10th hp mesh for the E-plane 90◦ bend with round corner

41

(a) |Ey|

(b) |Ex|

Figure 33. Magnitudes of Ey and Ex corresponding to the E-plane 90◦

42

-1

-0.5

0

0.5

1

1.5

2

900 1191 1540 1951 2430 2980 3608 4319 5118

log

(rel

ativ

e er

ror

%)

Ndof

E-plane 90 degrees bend with round corner

energy error

Figure 34. Convergence history for the E-plane 90◦ bend with round corner

43

3.2.3 E-plane capacitive symmetric iris

This discontinuity (Fig. 35) is due to a symmetric iris (as the one of Sec-tion 3.1.3), but in the E-plane. Thus, the FEM domain is identical to the oneused for the H-plane inductive symmetric iris, but with different boundaryconditions on the conductors boundaries (of Dirichlet type for this case). Theanalysis is made by exciting port 1 (on the left).

The initial mesh is shown in Fig. 36(a). Fig. 36(b) shows a sample meshcorresponding to the 4th iteration. A refinement pattern around the corners ofthe iris is observed due to the presence of field singularities at those locations.The convergence history (up to an error as low as 0.1%) is shown in Fig. 37.Exponential convergence is again observed.

Plots of the field component magnitudes |Ey| and |Ex| are shown in Fig. 38.A stationary wave pattern is observed in the input waveguide (between theexcitation port and the iris) as well as a singular behavior of the field at thecorners of the iris. The y-component of the field in the structure correspondsto the ∓η component of the waveguide modes. Analogously, the x-componentcorresponds to the ±ζ component of the waveguide. Thus, the Ex componentof the field is generated at the discontinuity, and it is only significant close toit.

The results for S11 and S21 corresponding to some of the iterations of the hpadaptivity are shown in Tab. 8. The equalities S21 = S12 and S22 = S11 holddue to reciprocity and symmetry of the structure. Only the results of the firstiterations are shown as the hp FEM results for the consecutive meshes arepresumed to be more accurate than those of MM.

Table 8Scattering parameters for the E-plane capacitive symmetric iris

|S11| |S21| arg(S11) arg(S21)

Iter. 1 0.3180 0.9481 -163.31◦ -53.24◦

Iter. 2 0.3070 0.9517 -162.71◦ -52.60◦

Iter. 5 0.3013 0.9535 -162.37◦ -52.25◦

Iter. 6 0.3009 0.9537 -162.35◦ -52.22◦

MM 0.3008 0.9537 -162.36◦ -52.23◦

44

y

x

Ω

Γ2pΓ1p

ε0, µ0

l1 t l2

E-plane

d

ΓD

ΓD

b

b

a

l

l2

l1

t ε0, µ0

ζ

ξ

ζ

ξη

η

ηζ

ζη

y

E-plane

x

Port 1

Port 2

Figure 35. E-plane capacitive symmetric iris (d/b = 0.6, t/b = 0.2)

45

48 49

52 53

54 55

62 63

60

61

56 57

58 59

64 65

50 51

x

y

z

(a) Initial mesh

x

y

z

(b) Mesh of 4th iteration

Figure 36. Initial mesh and mesh of the 4th iteration for the E-plane capacitivesymmetric iris

46

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

88437147568344343383251218061246816

log

(rel

ativ

e er

ror %

)

Ndof

E-plane capacitive symmetric iris

energy error

Figure 37. Convergence history for the E-plane capacitive symmetric iris

47

(a) |Ey|

(b) |Ex|

Figure 38. Magnitude of Ey and Ex corresponding to the E-plane capacitive sym-metric iris

48

3.2.4 E-plane double stub section

The structure is shown in Fig. 39. It consists of a main waveguide going fromport 1 to port 2 and two waveguides loading it (referred to as stubs). Thestubs are closed at their ends causing a total reflection of the energy at theirinputs. Thus, the load impedance that they present to the main waveguideis purely imaginary. The behavior of this structure may be roughly explainedas follows. The load of each stub is like a discontinuity in the waveguide,producing a reflected wave (and also a transmitted wave) with a given phase.As there are two stubs, i.e., two discontinuities, the contributions from thetwo stubs may (totally or partially) add or cancel, depending on the relativephase between the corresponding waves. The relative phase depends (for givenstubs dimensions) on the electrical distance θd between them. As the electricaldistance depends on the frequency for a given physical distance d, i.e., θd =β10d, the frequency response can be adjusted for several applications. Forexample, the double stub section can be designed to work as a phase shifter,i.e., causing an extra shift in the phase of the wave at the transmitted port for agiven frequency band. This is done by designing the double stub in such a waythat there is an adding interference at the transmitted port of the two wavesgenerated by the stubs (with a given phase). Another usual application is animpedance matching network, i.e., the double stub is designed to compensatethe reflection present at a given port, due to, e.g., a change in the height of thewaveguide, in a given frequency band. This is done by adjusting the design sothere is a cancellation of the two reflected waves at the stubs junctions (180◦

out of phase with respect to each other).

In here, the double stub has been designed for the latter application, i.e., tohave a null reflection around a frequency given by k0 = 1.39kc (kc being thecut-off frequency of the TE10 mode). Notice in Fig. 39 that the waveguidesections at the two ports are identical. Thus, if the two stubs were not presentthere would be no reflection (S11 ideally null) as the structure would simplyconsists of a waveguide section. The reason we choose to analyze the structurein Fig. 39 (which has a little practical application) is because this is a goodtest case. The idea is that for the null reflection frequency, the fields in themain waveguide have to be basically the same as those in a single waveguidesection. Since the stubs are identical, the structure is symmetrical (S11 = S22).As in the other cases, the analysis is made by exciting port 1 (on the left).

The frequency response of the structure around the frequency correspondingto k = 1.39Kc is shown in Fig. 40(a). The ordinate axis corresponds to S11in dB, i.e., 10 log10 |S11|2 7 . The results have been obtained by running the

7 The dB is a logarithmic unit for dimensionless magnitudes, but it is always withrespect to the power (and not the field) magnitudes. Thus, when applied to thescattering parameters that relate field quantities, the “10 log10” factor has to beapplied to the square of the scattering coefficient. For example, S11 = −40dB means

49

hp-adaptivity until an energy error of 1% is achieved and computing the S11using the final hp-mesh. The expected low values of the reflection coefficientS11 around k0/kc = 1.39 is observed. For a comparison, Fig. 40 presents alsoresults obtained using a MM technique. A very good agreement is observed.

Figure 40(b) shows the frequency response over a broad frequency interval thatconsists of a centered band covering the 60% of the monomode frequency band.A very good agreement between the hp-FEM and MM results is observed. Theonly exception is around the frequency corresponding to k0/kc = 1.74. Forthese frequencies, a more refined mesh seems to be needed.

The convergence has been studied by running the hp-adaptivity with anenergy-norm error tolerance of 0.01% (and a maximum number of iterationsequal to 20). Results corresponding to five significant frequency points (sym-metrically chosen around the value of k0/kc = 1.39): k0/kc=1.32, 1.36, 1.42,1.46 are displayed in Fig. 41. For k0/kc = 1.32, 1.46 there is a high reflection ofthe energy at the input waveguide; for k0/kc = 1.39 there is a very low (almostnull) reflection at the input; and for k0/kc = 1.36, 1.42 an intermediate situa-tion occurs. Except for the first few iterations, the plots follow approximatelya straight line, reflecting an exponential decrease of the energy-norm error.The erratic behavior of the error during the first few iterations is due to thecoarseness of the initial mesh (shown in Fig. 42(a)).

The final meshes for k0/kc = 1.32 (high reflection at the input), k0/kc = 1.39(low reflection at the input), and k0/kc = 1.36 (intermediate reflection at theinput) are displayed in Figures 42 and 43. For the case of high reflection, themesh (shown in Fig. 42(b)) displays a typical refinement pattern around thecorners (junctions of the stubs with the main waveguide). The electric field 8

for this case is plotted in Fig. 44(a). A stationary wave pattern in the inputwaveguide is observed due to the interference between the incident and thereflected waves. On the other hand, the mesh for the low reflection case (shownin Fig. 43(b)) displays a situation very similar to the situation of the smoothfield solution inside a waveguide section (i.e., without singularities). As it wasexplained above, this occurs because the stubs do not load the main waveguide(it is like if they were not present in the structure). This is best understood byseeing Fig. 44(b), which displays the electric field in the structure for this case.The stationary wave pattern at the input waveguide can hardly be seen, whichmeans that the level of the reflected wave is very low. Finally, the mesh forthe intermediate case (k0/kc = 1.36) is shown in Fig. 43(a). It is observed how

that the power reflected at the input waveguide is 40 dB below the power of theexcitation at the input waveguide, i.e., 104 lower (or equivalently, the electric andmagnetic fields are 102 lower).8 Note that the magnitude plotted is

√|Ex|2 + |Ey|2 that, although does not cor-

respond to |E| or other physically meaningful magnitude, it is useful for visualizingin one plot the field in the main waveguide and in the stubs.

50

effectively the mesh corresponds to an intermediate case between the meshesof Figures 42(b) and 43(b).

xy

η

ξζ

Port 1

l1

ξ

ζ

η

a

b

b

b

l2

bs1

bs2

l

E-plane

Port 2

y

x

Ω

Γ1

p

b

ζ

η

b b

bε0, µ0

bs1 bs2

Γ2

p

ΓD

ΓD

E-plane

ζ

ηΓD

ΓDΓD

ΓD

ΓD ΓD

l

l1 l2

Figure 39. E-plane double stub structure (bs1/b = bs2/b = 5.0249, l/b = 1.2608)

51

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

1.3 1.35 1.4 1.45 1.5

dB

K0/Kc

E-plane double stub

|S11||S11| (MM)

(a) Frequency response in the band of interest

-50

-40

-30

-20

-10

0

1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8

dB

K0/Kc

E-plane double stub

|S11||S11| (MM)

(b) Frequency response over the 60% of the monomode region

Figure 40. Frequency response of E-plane double stub section (|S11| in dB)

52

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

571 1067 1790 2783 4087 5747 7804 10301 13281 16786 20860

log

(rel

ativ

e er

ror

%)

Ndof

E-plane double stub

energy error (K0/KC=1.32)energy error (K0/KC=1.36)energy error (K0/KC=1.39)energy error (K0/KC=1.42)energy error (K0/KC=1.46)

Figure 41. Convergence history for the E-plane double stub section

53

59 60 61 62

63

64

65

66

67 68 69

70

71

72

73

74 75 76 77

x

y

z

(a) Initial mesh

59 60 61

78 79

102

103

176

177

334

335

412

413

508

509

510

511

415

337

179

105

81

592

593

594

595

519

520

521

345

346

347

231

232

233

209

210

211

274

486

570

571

572

573

488

489

276

277

154

155

64

65

66

90 91

186

187

288

289

498

499

582

583

584

585

501

291

189

166

167

168

220

221

222

354

355

356

422

423

424

530

531

532

533

112 113

242

243

432

433

604

605

606

607

435

245

196

197

198

254

255

256

442

443

444

540

541

542

543

126 127

312

313

402

403

550

551

552

553

405

315

322

323

324

452

453

454

614

615

616

617

624

625

626

627

463

464

465

367

368

369

378

474

636

637

638

639

476

477

380

381

300

301

71

72

73

138 139

140

264

265

266

392

393

394

558

559

560

561

75 76 77

x

y

z

(b) Mesh for K0/Kc = 1.32

Figure 42. Initial mesh and mesh corresponding to 1% energy error for the E-planedouble stub section

54

59 60 61

78 79

142

143

256

257

258

259

145

81

354

355

356

357

309

310

311

286

366

367

368

369

288

289

186

187

64

65

66

90 91

198

199

298

299

300

301

201

208

209

210

322

323

324

325

102 103

152

153

378

379

380

381

155

162

163

164

266

267

268

269

116 117

234

235

344

345

346

347

237

244

245

246

414

415

416

417

400

401

402

403

332

388

389

390

391

334

335

222

223

71

72

73

128 129

130

174

175

176

276

277

278

279

75 76 77

x

y

z

(a) Mesh for K0/Kc = 1.36

59 60 61

78 79

80

81

63

64

65

66

90 91

92

93

102 103

104

105

116 117

118

119

70

71

72

73

128 129

130

131

75 76 77

x

y

z

(b) Mesh for K0/Kc = 1.39

Figure 43. Meshes corresponding to 1% energy error for the E-plane double stubsection

55

(a) High reflection at the input waveguide (K0/Kc = 1.32)

(b) Low reflection at the input waveguide (K0/Kc = 1.39)

Figure 44. Electric field√|Ex|2 + |Ey|2 in the E-plane double stub section

56

4 Conclusions

An hp-adaptive Finite Element Method for studying the characterization ofmicrowave rectangular waveguide discontinuities with a geometry invariantalong one direction (a common situation in rectangular waveguide technology),has been presented. The assumption on the geometry of the discontinuityenables a 2D analysis in so called H-plane or E-plane of the structure.

A fully automatic hp-adaptive strategy based on maximizing the rate of de-crease of the (projection-based) interpolation error of the fine grid solution,has been applied to a number of important engineering examples. Computa-tion of the scattering matrix that characterize the electromagnetic behaviorof the discontinuities for the microwave engineer has been implemented as apost-processing of the solution.

A wide variety of structures have been analyzed, including microwave engi-neering devices of medium complexity. The hp adaptivity has shown to deliverexponential convergence rates for the error for both regular and singular solu-tions. A consistent convergence pattern makes us believe that the results aremore accurate than those obtained with semi-analytical techniques. At thesame time, this hp-methodology presents the important advantage of beinga purely numerical method, which allows for modeling of complex waveguidestructures that cannot be solved by using semi-analytical techniques.

5 Acknowledgment

The authors would like to thank Sergio Llorente-Romano at the UniversidadPolitécnica de Madrid for their helpful discussions on the MM techniques andfor letting us use their MM codes that have been used to produce some of theMM results shown in this paper.

57

References

[1] L. E. Garćıa-Castillo, D. Pardo, L. F. Demkowicz, A two-dimensionalsel-adaptive hp-adaptive finite element method for the characterization ofwaveguide discontinuities. Part I: Waveguide theory and finite elementformulation(Submitted to Computer Methods in Applied Mechanics andEngineering).

[2] N. Marcuvitz, Waveguide Handbook, IEE, 1985.

[3] J. Uher, J. Bornemann, U. Rosenberg, Waveguide Components for AntennaFeed Systems: Theory and CAD, Artech House Publishers, Inc., 1993.

[4] L. F. Demkowicz, W. Rachowicz, P. Devloo, A fully automatic hp-adaptivity,Journal of Scientific Computing 17 (1-3) (2002) 127–155.

[5] L. Demkowicz, Computing with hp Finite Elements. I. One- and Two-Dimensional Elliptic and Maxwell Problems, CRC Press, Taylor and Francis,2006.

[6] W. Rachowicz, L. F. Demkowicz, An hp-adaptive finite element methodfor electromagnetics. Part 1: Data structure and constrained approximation,Computer Methods in Applied Mechanics and Engineering 187 (1-2) (2000)307–337.

[7] L. Vardapetyan, L. F. Demkowicz, Full-wave analysis of dielectric waveguidesat a given frequency, Mathematics of Computation 75 (2002) 105–129.

[8] L. Vardapetyan, L. F. Demkowicz, D. Neikirk, hp vector finite element methodfor eigenmode analysis of waveguides, Computer Methods in Applied Mechanicsand Engineering 192 (1-2) (2003) 185–201.

[9] L. Vardapetyan, L. Demkowicz, hp-adaptive finite elements in electromagnetics,Computer Methods in Applied Mechanics and Engineering 169 (3-4) (1999)331–344.

[10] W. Rachowicz, L. F. Demkowicz, A two-dimensional hp-adaptive finite elementpackage for electromagnetics (2Dhp90 EM), Tech. Rep. 16, Institute forComputational Engineering and Sciences (1998).

[11] W. Rachowicz, L. Demkowicz, An hp-adaptive finite element method forelectromagnetics. Part 2: A 3D implementation, International Journal forNumerical Methods in Engineering 53 (1) (2002) 147–180.

[12] L. F. Demkowicz, Fully automatic hp-adaptivity for Maxwell’s equations,Computer Methods in Applied Mechanics and Engineering 194 (2-5) (2005)605–624, see also ICES Report 03-45.

[13] F. Ihlenburg, I. Babuška, Finite element solution of the Helmholtz equation withhigh wave number. I: The h-version of the FEM, Computer and Mathematicswith Applications 30 (9) (1995) 9–37.

58

[14] F. Ihlenburg, I. Babuška, Finite element solution of the Helmholtz equationwith high wave number. II: The h − p-version of the FEM, SIAM Journal ofNumerical Analysis 34 (1) (1997) 315–358.

[15] A. Wexler, Solution of waveguide discontinuities by modal analisis, IEEETransactions on Microwave Theory and Techniques 15 (1967) 508–517.

[16] T. Itoh, Numerical Techniques for Microwave and Millimeter Wave PassiveStructures, John Wiley & Sons, Inc., 1989.

[17] D. Pardo, L. E. Garćıa-Castillo, L. F. Demkowicz, A two-dimensionalself-adaptive hp-adaptive finite element method for the characterization ofwaveguide discontinuities. Part III: Goal-oriented hp-adaptivity(Submitted toComputer Methods in Applied Mechanics and Engineering).

[18] D. Pardo, L. Demkowicz, C. Torres-Verdin, L. Tabarovsky, A goal orientedhp-adaptive finite element method with electromagnetic applications. Part I:Electrostatics, Tech. Rep. 57, ICES, to appear at International Journal forNumerical Methods in Engineering (2004).

[19] D. Pardo, L. Demkowicz, C. Torres-Verdin, M. Paszynski, Simulation ofresistivity logging-while-drilling (LWD) measurements using a self-adaptivegoal-oriented hp-finite element method(Submitted to SIAM Journal on AppliedMathematics).

[20] M. Paszynski, L. Demkowicz, D. Pardo, Verification of goal-oriented hp-adaptivity, Computer and Mathematics with Applications 50 (6) (2005) 1395–1404, see also ICES Report 05-06.

[21] L. F. Demkowicz, Encyclopedia of Computational Mechanics, John Wiley &Sons, Inc., 2004, Ch. “Finite Element Methods for Maxwell Equations”.

[22] W. Gui, I. Babuška, The h, p and h−p versions of the finite element method in 1dimension - Part I. The error analysis of the p-version, Numerische Mathematik49 (1986) 577–612.

[23] W. Gui, I. Babuška, The h, p and h−p versions of the finite element method in 1dimension - Part II. The error analysis of the h- and h−p versions, NumerischeMathematik 49 (1986) 613–657.

[24] W. Gui, I. Babuška, The h, p and h − p versions of the finite element methodin 1 dimension - Part III. The adaptive h− p version, Numerische Mathematik49 (1986) 659–683.

[25] I. Babuška, B. Guo, Regularity of the solutions of elliptic problems withpiecewise analytic data. Part I. Boundary value problems for linear ellipticequation of second order, SIAM Journal of Mathematical Analysis 19 (1) (1988)172–203.

[26] I. Babuška, B. Guo, Regularity of the solution of elliptic problems withpiecewise analytic data. II: The trace spaces and application to the boundaryvalue problems with nonhomogeneous boundary conditions, SIAM Journal ofMathematical Analysis 20 (4) (1989) 763–781.

59

[27] I. Babuška, B. Guo, Approximation properties of the hp-version of the finiteelement method, Computer Methods in Applied Mechanics and Engineering133 (1996) 319–346.

[28] C. Schwab, p- and hp- Finite Element Methods. Theory and Applications inSolid and Fluid Mechanics, Oxford University Press, 1998.

[29] W. Rachowicz, D. Pardo, L. F. Demkowicz, Fully automatic hp-adaptivity inthree dimensions, Tech. Rep. 22, ICES, to appear at Computer Methods inApplied Mechanics and Engineering (2004).

[30] D. Pardo, L. F. Demkowicz, Integration of hp-adaptivity and multigrid. I. Atwo grid solver for hp finite elements, Computer Methods in Applied Mechanicsand Engineering 195 (7-8) (2005) 674–710, see also ICES Report 02-33.

[31] D. Pardo, Integration of hp-adaptivity with a two grid solver: Applications toelectromagnetics, Ph.D. thesis, University of Texas at Austin (2004).

[32] L. F. Demkowicz, P. Monk, L. Vardapetyan, W. Rachowicz, De Rham diagramfor hp finite element spaces, Computer and Mathematics with Applications39 (7-8) (2000) 29–38.

[33] L. F. Demkowicz, I. Babuška, Optimal p interpolation error estimates for edgefinite elements of variable order in 2D, SIAM Journal of Numerical Analysis41 (4) (2003) 1195–1208.

[34] L. F. Demkowicz, A. Buffa, H1, H(curl) and H(div)-conforming projection-based interpolation in three dimensions. Quasi optimal p-interpolationestimates, Computer Methods in Applied Mechanics and Engineering 194 (24)(2005) 267–296, see also ICES Report 04-24.

[35] L. F. Demkowicz, Projection-based interpolation, no. 03, Cracow Universityof Technology Publications, Cracow, 2004, monograph 302, A special issue inhonor of 70th Birthday of Prof. Gwidon Szefer, see also ICES Report 04-03.

60

a two-dimensional self-adaptive hp finite element method ...waveguide discontinuities, i.e., the...

Documents